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\title{On the Cauchy Problem for
  non-effectively hyperbolic operators, the Gevrey 6 well-posedness}
\author{Enrico Bernardi
\thanks{Dipartimento di Matematica Per Le Scienze
Economiche e Sociali, Universit\`a di Bologna, Viale Filopanti 5,
40126 Bologna, Italy}
}



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\begin{document}
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%\section{}
%\subsection{}

\begin{abstract}
In this paper we prove an optimality result for the well-posedness of
the Cauchy problem for some hyperbolic operators with double
characteristics in the realm of Gevrey classes. The class of operators
considered here allows us to complete the analysis of this type of
problem in a suitably stable geometric setting and to understand the
fine and subtle interplay between the behaviour of the Hamilton flow
and the role of the lower order terms satisfying the necessary
Ivrii-Petkov conditions.
\end{abstract}
\section{Introduction}
In this paper we prove the following result: let $ P $ be the linear partial differential operator
\begin{equation}
  \label{eq:110}
  P(x,D) = -D_{0}^{2} + 2x_{1}D_{0}D_{n} + D_{1}^{2}+
  x_{1}^{3}D_{n}^{2} + D_{2}^{2} + x_{2}^{2}D_{n}^{2} + A D_{n},
\end{equation}

and assume that the necessary (microlocal) Ivrii-Petkov
condition for well-posedness is satisfied at the double point $
\rho = (0,e_{n}) $, i.e.  $ \text{Tr}^{+}F(\rho) + p^{s}_{1}(\rho) =
A + 1 > 0 $.

Then:
\begin{theorem}
\label{thm}
If $s>6$ then the Cauchy problem for $P$ is not locally solvable at the origin in the Gevrey $s$ class.
\end{theorem}


% In this paper we continue our study begun in \cite{BeNi} of hyperbolic
% operators with double characteristics and Hamilton flow touching the
% double manifold. Our motivation here is similar to \cite{NC} where a
% result was proved which showed that if lower order terms were added to a model operator and
% these terms did not verify the necessary Ivrii-Petkov conditions for
% the $ C^{\infty} $ well-posedness, in the particular geometrical
% setting which we shall present in a moment, then one could not possibly
% go beyond the Gevrey threshold of $ s = 6 $ when well-posedness of the
% Cauchy problem is concerned. Here we will assume those conditions
% verified and prove that still the  class $ \gamma^{6} $ is optimal.

% Our operators belong to the class of hyperbolic symbols $ p(x,\xi) $
% in $ x = (x_{0},x') \in \R^{n+1} $ vanishing of order $ 2 $ on a
% smooth sub-manifold $ \Sigma_{2} \subset T^{*}\R^{n+1} $ on which the
% canonical symplectic 2-form $
% \sigma $ has constant rank. We shall
% assume that it is possible to find a null bicharacteristic issued from a
% simple point and landing tangentially onto $ \Sigma_{2} $. Under some
% generic geometrical stability conditions, it was proved in \cite{BBP}
% and \cite{N3} that this amounts to the non-vanishing of a certain term in
% the third order Taylor development of the principal symbol $
% p_{2}(x,\xi) $ near $ \Sigma_{2} $. This class of operators, of which
% (\ref{eq:110}) and (\ref{eq:10}) below are microlocal models,  was
% not included in the very general treatment of the Cauchy problem presented
% \cite{Iv1}  and \cite{Ho1}, precisely because of this
% geometrical feature. The first positive result in a model case
% exhibiting this behaviour was given in \cite{BB2}
% where it was proved that the Cauchy problem could be proved to be
% well-posed in the Gevrey class $ \gamma^{s} $ up to $ s = 5 $ or $ s
% =6 $, depending whether the subprincipal symbol were identically $ 0 $
% or else satisfied a strictly positive trace condition. In \cite{BeNi}
% we were able to  generalize this result to a general class of operators and
% furthermore, we proved that then $ s = 5 $ is the optimal Gevrey threshold. In
% \cite{NP} the same result was extended when the lower order terms
% verify the Ivrii-Petkov conditions, thus concluding well-posedness up
% to Gevrey $ 6 $.

% The operators we are concerned with here are non-effectively
% hyperbolic, i.e. $ \forall \rho \in \Sigma_{2} $ we have $
% \text{Sp}F_{p_{2}}(\rho) \subset i\R $, where $ F_{p_{2}}(\rho) $ is
% the fundamental matrix of the principal symbol evaluated at the double
% point $ \rho $. We are further assuming that $ 0 $ is an eigenvalue of
% $ F_{p_{2}}(\rho) $ at which the Jordan decomposition of $
% F_{p_{2}}(\rho) $ has a block of dimension 4, the highest possible
% according to the classical mechanics classification of hyperbolic
% symplectic quadratic forms, see \cite{Ho2} or \cite{Ho1}. H\"ormander
% in \cite{Ho3} was already aware that the presence of this higher order
% Jordan blocks could reverberate on the generic Gevrey regularity,
% which due to a general result of Bronstein \cite{Bro} never exceeds $
% s=2 $ for a generic hyperbolic operator with double
% characteristics. In \cite{Ho3} it was proved, albeit on a model case
% and using an explicit fundamental solution, that the Gevrey threshold
% moves up to $ s=4 $. This result has been recovered in \cite{NC}
% always in a model case, but using a weighted energy estimates approach
% suitable to generalizations which will appear in \cite{BeNi4}. The
% typical model of such a situation would be:%
% \begin{equation}
% \label{mo1}
%  P(x,D) = -D_{0}^{2} + 2D_{0}D_{1} + x_{1}^{2}D_{n}^{2} ;
% \end{equation}
% %
% if we add to $ P $ in (\ref{mo1}) $ bD_{n} $ with a generic complex $
% b $ we don't go beyond Gevrey $ 4 $, whereas if $ b > 0 $
% (microlocally near $ (0,e_{n}) $) we jump to $ C^{\infty} $ well-posedness.

% The situation changes completely for the operators considered in this
% paper, for example when studying as in \cite{BeNi}:

% \begin{equation}
%   \label{mo2}
%   P(x,D) = -D_{0}^{2} + 2x_{1}D_{0}D_{n} + D_{1}^{2}+
%   x_{1}^{3}D_{n}^{2},
% \end{equation}
% we have at most Gevrey well-posedness up to $ s =5 $ as already
% mentioned  and, this is the main
% result of the present  paper, even allowing for a well-oriented subprincipal
% symbol, i.e. adding $ bD_{n} $ with $ b > 0 $, we don't go beyond
% Gevrey $ s = 6 $. So we see that the presence of null
% bicharacteristics falling on the double manifold, which do exist for
% model (\ref{mo2}) prevents the recovering effect of total
% well-posedness that a good lower order term would otherwise induce and
% still hinders $ C^{\infty} $ well-posedness, limiting the Gevrey
% order: nevertheless the Ivrii-Petkov conditions are the cause of an
% improvement in the Gevrey regularity:
% we progress a little from $ s = 5 $ to $ s = 6 $.

% The idea of the proof here is the same as in \cite{BeNi}: we want to
% find a null solution of the operator violating an apriori estimate
% holding whenever the Gevrey Cauchy problem is well-posed. The ordinary
% differential equation to which the search of a null solution reduces
% still belongs to the Sibuya class thoroughly studied in \cite{Si}.

% In \cite{BeNi} we were able to find a good solution solving a Stokes
% coefficient equation,using a result first obtained in \cite{BB1} in
% the $ C^{\infty} $ case, thus connecting two separate asymptotic
% expansions at $ -\infty $ and $ +\infty $. The problem here is more
% difficult since now the Stokes coefficient, due to the presence of
% lower order terms, is a function of two variables, so there are no
% easily applicable root-finding mechanisms.
% We should mention that the problem of finding zeroes of this type of
% Stokes coefficient for this and similar ordinary differential
% equations has been much investigated, due also to the $ {\cal PT}
% $-symmetry these types of equations exhibit, see
% e.g. \cite{DT},\cite{Shin1},\cite{Shin2}. This problem is easily seen
% to be equivalent to that of the study of the eigenvalues of equations
% like (\ref{mo4}) below.
% We see that after Fourier transforming and  scaling (\ref{mo2}) we are
% left with the ordinary differential equation:
% %
% \begin{equation}
% \label{mo3}
% u''(x) = (x^{3} + a_{2}x + a_{3})u(x),
% \end{equation}
% %
% a so called anharmonic cubic oscillator equation. Starting with the
% pioneering work of Bender and Wu \cite{BW} in the case of quartic
% anharmonic oscillators the properties of analyticity of the
% eigenvalues of equations like (\ref{mo3}) and their quartic analogous
% have been much studied. A recent approach encompassing previous known
% results is contained in \cite{EG} where for the $ {\cal PT}
% $-symmetric cubic oscillator %
% \begin{equation}
% \label{mo4}
% -y'' + (iz^{3} + i\alpha z)y = \lambda y,
% \end{equation}
% %
% with boundary conditions $ y(z) \rightarrow 0 $ as $ z \in \R $, $ z
% \rightarrow \pm \infty $ it is stated that the set of zeroes of the
% corresponding Stokes coefficient is an irreducible complex
% non-singular curve. While powerful this result, it is not enough in our
% case, since we need to control the intersections of the zero set with
% the positive real axis in $ a_{3} $. Thus we resorted to a less
% ambitious program: instead of looking for exact eigenvalues, we
% estimated the solutions and their related Wronskians determinants when
% $ a_{3} $ is large and $ a_{2} $ too in the homogeneous scaling $
% a_{2} \sim a_{3}^{2/3} $, in order to quasi-connect the asymptotic
% expansions of the Sibuya solutions modulo factors behaving like $
% e^{-|a_{3}|^{\kappa}} $, with $ \kappa $ related to the Gevrey order
% of the problem.
% This turned out to be easier and eventually allowed us
% to prove our optimality result.
% In fact the introduction here of terms purely imaginary in the
% exponentials of the null solutions, the $ \alpha $ in (\ref{de}) below, absent in the solution of
% \cite{BeNi} and  thus invisible to the estimates,
% plays nevertheless a very important role inasmuch as it yields an
% exponentially decreasing force to some Wronskian determinants. These
% terms take into account the fact the if we were to solve the Hamilton
% system with the lower order terms of the right homogeneity included,
% we would have a bending of the null bicharacteristics previously falling onto $
% \Sigma_{2} $, the strength of which finally explains the upper Gevrey
% bound shifting upwards from $ s = 5 $ to $ s =6 $.

Let us recall that we say that $f(x)\in C^{\infty}({\mathbb R}^n)$
belongs  to $\gamma^{(s)}({\mathbb R}^n)$, the Gevrey space of order
$s$,  where $ s \geq 1 $,  if for any compact set $K \subset {\mathbb
  R}^n$  there exist $C>0$, $h>0$ such that
%
\begin{equation}
\label{Gevreyineq}
\bigl|\dif^{\alpha}_xf(x)\bigr|\leq Ch^{|\alpha|}|\alpha|!^s,\quad x\in K
\end{equation}
%
for every $\alpha\in {\mathbb N}^n$. In particular $\gamma^{(1)}({\mathbb R}^n)$ is the space of real analytic functions on ${\mathbb R}^n$.


%\section{The symbol}



The paper is organized as follows: in section \textbf{\ref{sec3}} we
prove, collect and organize
all the asymptotic results on the Stokes coefficients of equation
(\ref{de}) while briefly recalling the theory of Sibuya's subdominant solutions
and Stokes coefficients asymptotics.
In section \textbf{\ref{opt}} we prove Theorem \ref{thm} while recalling some
technical tools from \cite{BeNi} to handle Gevrey estimates.


\section{Asymptotic Analysis for $ x $ and $ \lambda $ large}
%\section{The Solution}
\label{sec3}
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%\setcounter{exercise}{0}


\subsection{Preliminary Lemmas}
Denote by $ \arg z $ the principal value of the argument of the
complex number $ z $, $ -\pi < \arg z \leqq \pi $.
%
\begin{lemma}
\label{lemma1}
Let $ 1>  \rho_{0} > 0 $ , $ x \geqq 0 $, $ y > 0 $ and $ 0 \leqq \theta
\leqq \pi - \rho_{0} $. Then we have
\[
 0 \leqq  \arg (x + y
e^{i\theta}) \leqq \theta .
\]
\end{lemma}
%
%
\begin{proof}
 Recall that with $ u \in \R, v \in \R, u \geqq 0, v >  0 $%
\begin{equation*}
%\label{}
\arg ( u + i v) = \frac{1}{2i}(\log(u + iv) - \log(u - iv))
\end{equation*}
%
with the principal determination of the complex logarithm chosen.

Put

\[
\varphi(x) =  \arg(x + y e^{i\theta}) = \frac{1}{2i}[\log ( x + y
e^{i\theta}) - \log( x + y e^{-i\theta})]
\]

We have $ \phi(0) = \theta $ and, assuming $ \theta \in ]0,\pi -
\rho_{0}] $ we have:
\[
\phi'(x) = - \frac{2y\sin\theta}{x^{2} + y^{2} + 2xy\cos\theta} < 0
\]
we get the result.
\end{proof}
%
\begin{lemma}
\label{lemma2}
Let  $ x \geqq 0 $, $ y > 0 $ and $ | \psi | < \frac{5\pi}{6} $. Then
there exists a positive $ \kappa $ such that
\[
 | x + y e^{i\psi}| \geqq \kappa (x + y) .
\]
\end{lemma}

\begin{proof}
 Clear from $ | x + y e^{i\psi}|^{2} > x^{2} + y^{2} -\sqrt{3}xy $.
\end{proof}
In the following $ \lambda $ will be a large complex parameter $ \lambda \in \C $ , $ |\lambda| \gg 1 $ with $ | \arg
\lambda | \leqq \pi - \rho_{0} $.

We are interested in %
\begin{equation}
\label{p1}
p(x,\lambda) = x^{3} + (A\lambda^{4\theta} + i b \lambda^{-\nu})x + B
\lambda = p_{0}(x,\lambda) + i b \lambda^{-\nu}x,
\end{equation}
%
with $ A, B > 0 $, $ b \in \R $, $ \frac{1}{6} < \theta $, $ \nu > 0
$.

We shall start proving the following for $ p_{0}(x,\lambda) $:

\begin{lemma}
Assume that $ |\arg \lambda | \leqq \pi - \rho_{0} $ and $ x \geqq 0 $,
with $ \rho_{0} $ is an arbitrary but fixed sufficently small positive
constant. Then there exist two positive numbers $ r_{0}, M_{0} $
depending  only on $ \rho_{0}, A, B $ such that:
\[
\begin{cases}
  |p_{0}(x,\lambda)| \geqq r_{0}x^{3} \\
  |p_{0}(x,\lambda)| \geqq r_{0}|\lambda| \\
  |\arg p_{0}(x,\lambda) | \leqq \pi - \frac{\rho_{0}}{2}\\

\end{cases}
\]
\end{lemma}
\subsection{The Subdominant Solution}

The differential equation is:
%
\begin{equation}
\label{mainde}
y''(x) = p(x,\lambda)y = (x^{3} + (A\lambda^{4\theta} + i b \lambda^{-\nu})x + B
\lambda)y
\end{equation}
%
with the same conditions on $ A,B,b, \lambda, \theta $ as in (\ref{p1}).

In order to prove $u(\lam^2 x)={\mathcal Y}_0(\lam^2x;\lam^{2/3}a(\lam),\lam b(\lam))$
is indeed in some Gevrey class for $|x|<1$ and for large $\lam$, we now study how ${\mathcal Y}_0(x;a\lam^{2/3},b\lam )$ behaves for large $|x|$ and large $|\lam|$ where $|a|, |b| \leq M$.
%
\begin{proposition}
\label{pro:esti:Y}
We have
%
\begin{eqnarray*}
&&{\mathcal Y}_0(x;a\lam^{2/3},b\lambda)\\
&&=(1+p(x,\lambda))e^{{\hat E}(x,\lambda)}x^{-3/4}\exp{\{-\bigl(\frac{2}{5}x^{5/2}+a\lam^{2/3} x^{1/2}+r(x,\lam)\bigr)\}},\\
&&{\mathcal Y}_0'(x;a\lam^{2/3},b\lambda)\\
&&=(-1+p(x,\lambda))e^{{\hat E}(x,\lambda)}x^{3/4}\exp{\{-\bigl(\frac{2}{5}x^{5/2}+a\lam^{2/3} x^{1/2}+r(x,\lam)\bigr)\}}
\end{eqnarray*}
%
in
%
\[
S_{\lam}=\{x;|\arg x|<\frac{3\pi}{5}, |x|> |\lambda|^{4/7}, |\lambda|> R\}
\]
%
for $|a|, |b|\leq M$ where
%
\[
r(x,\lam)=-b\lam x^{-1/2}+\frac{a^2}{12}\lam^{4/3}x^{-3/2}+\frac{ab}{10}\lam^{5/3} x^{-5/2}
\]
%
and $p(x,\lambda)$, ${\hat E}(x,\lambda)$ are holomorphic in $S_{\lam}$ and in any closed subsector of $S_{\lam}$ we have
%
\[
|p(x,\lambda)|\leq C|\lambda|^{-20/21},\quad |{\hat E}(x,\lambda)|\leq C
\]
%
and ${\hat E}(x,\lam)\to 0$, $p(x,\lambda)\to 0$  as $|x|\to\infty$, $x\in S_{\lam}$.
\end{proposition}
\subsection{The Subdominant Solution: Asymptotics in $\R^{+}$}

\begin{theorem}
 Assume that
\[
\begin{cases}
  |\arg \lambda| \leqq \pi - \rho_{0} \\
  0 \leqq x < +\infty \\
  0<A,B, |b| \leqq R_{0}\\
  \frac{1}{6} < \theta < \frac{1}{6} + \epsilon
\end{cases}
\]

where $ \epsilon,\rho_{0}, R_{0} $ are arbitrary but fixed positive
constants. Then, there exists  positive constants $ M_{0}, M_{1} $ such that,
if $ |\lambda| \geqq M_{0} $

\begin{eqnarray*}
&&{\mathcal Y}_0(x,A,B,b,\theta,\lambda)\\
&&=\widehat{C}(A,B,b,\theta,\lambda)(1+F_{1}(x,\lambda))p(x,\lambda)^{-\frac{1}{4}}
e^{-\int_{0}^{x}\sqrt{p(t,\lambda)}dt},\\
&&{\mathcal Y}'_0(x,A,B,b,\theta,\lambda)\\
&&=\widehat{C}(A,B,b,\theta,\lambda)(-1+F_{2}(x,\lambda))p(x,\lambda)^{-\frac{1}{4}}
e^{-\int_{0}^{x}\sqrt{p(t,\lambda)}dt},\\
\end{eqnarray*}
%

where
\begin{itemize}
\item $ p(x,\lambda) = x^{3} + (A\lambda^{4\theta} + i b \lambda^{-\nu})x + B\lambda  $
\item $ F_{1}(x,\lambda) $ and $ F_{2}(x,\lambda) $ tend to zero
  uniformly for
\[
0<A,B, |b| \leqq R_{0}, ~~ |\arg \lambda| \leqq \pi - \rho_{0}, ~~
|\lambda| \geqq M_{0}
\]
as $ x $ tends to $ +\infty $.
\item  $ F_{1}(x,\lambda) $ and $ F_{2}(x,\lambda) $ tend to zero
  uniformly for
\[
 0<A,B, |b| \leqq R_{0}, ~~ 0 \leqq x < +\infty
\]
as $ \lambda $ tends to infinity in the sector $ |\arg \lambda | \leqq
\pi - \rho_{0}$.
\item
\[
\widehat{C}(A,B,b,\theta,\lambda) = e^{-A^{5/4}K_{0}\lambda^{5\theta}\{ 1 +
  B\lambda^{1-6\theta}I_{1}(\lambda) + ib\lambda^{-\nu -4\theta}I_{2}(\lambda)\} }
\]

as $ \lambda $ tends to infinity in the sector $ |\arg \lambda | \leqq
\pi - \rho_{0}$ and $ |I_{j}(\lambda)| \leqq M_{1} $ when $ |\lambda|
\geqq M_{0} $, $ j=1,2 $, with $ K_{0} =
\frac{5\Gamma(-\frac{5}{4})^{2}}{64\sqrt{\pi}} \sim 0.6777$
\end{itemize}
\end{theorem}

\subsection{The Stokes Coefficient}


\section{Optimality}
\label{opt}

\subsection{Asymptotics for Large $x$}

\subsection{Asymptotics for Bounded $x$}

First estimate. We consider the equation:

%
\begin{equation}
\label{mainde2}
y''(x) = p(x,\lambda)y = (x^{3} + (A\lambda^{4\theta} + i b
\lambda^{-\nu})x + B\lambda + O(\lambda^{-2 +8\theta}))y
\end{equation}
%

in the region $ -c\lambda^{1 -4\theta} \leqq x \leqq 0 $, with $ c > \frac{B}{A} $.

\begin{lemma}

\end{lemma}
Second estimate. We consider the equation:

%
\begin{equation}
\label{mainde2}
y''(x) = p(x,\lambda)y = (x^{3} + (A\lambda^{4\theta} + i b
\lambda^{-\nu})x + B\lambda + O(\lambda^{-2 +8\theta}))y
\end{equation}
%

in the region $ -\lambda^{\frac{1}{3} + \psi(\epsilon)} \leqq x \leqq
-c\lambda^{1 -4\theta} $

\begin{lemma}

\end{lemma}
\subsection{The Proof of The Theorem}


\section{Sufficiency}


\newpage
\begin{thebibliography}{99}
%
\bibitem{Ahl}
{\sc L.Ahlfors}; Complex Analysis, MacGraw-Hill, 1979.
%
%
\bibitem{BW}
{\sc C.Bender and T.Wu}; {\it Anharmonic oscillators,} Phys.Rev.(2) {\bf 184} (1969), 1231-1260.
%
\bibitem{BB1}
{\sc E.Bernardi and A.Bove}; {\it On the Cauchy
problem for some hyperbolic operator with double
characteristics,} in Phase Space Analysis of Partial Differential
Equations,pp. 29-44, A.Bove {\it et al} eds.  Birkh\"auser, 2006
%
\bibitem{BB2}
{\sc E.Bernardi and A.Bove}; {\it A remark on the Cauchy Problem for a Model Hyperbolic Operator,}
in V.Ancona, J.Gaveau (ed.'s), Hyperbolic Differential Operators and Related Problems, 41-52, 2002. Marcel Dekker, New
York.
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